Existence of singular solutions of a degenerate equation in R
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Authors: | Shu-Yu Hsu |
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Institution: | (1) Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi, 621, Taiwan |
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Abstract: | Let a1,a2, . . . ,am ∈ ℝ2, 2≤f ∈ C(0,∞)), gi ∈ C(0,∞)) be such that 0≤gi(t)≤2 on 0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−ai|→−gi(t) as |x−ai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain. |
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Keywords: | Singular solution degenerate equation existence blow-up rate points of singularities |
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